Problem: Solve for $x$ : $ 3|x - 4| + 6 = 1|x - 4| + 10 $
Explanation: Subtract $ {1|x - 4|} $ from both sides: $ \begin{eqnarray} 3|x - 4| + 6 &=& 1|x - 4| + 10 \\ \\ { - 1|x - 4|} && { - 1|x - 4|} \\ \\ 2|x - 4| + 6 &=& 10 \end{eqnarray} $ Subtract ${6}$ from both sides: $ \begin{eqnarray} 2|x - 4| + 6 &=& 10 \\ \\ { - 6} &=& { - 6} \\ \\ 2|x - 4| &=& 4 \end{eqnarray} $ Divide both sides by ${2}$ $ \dfrac{2|x - 4|} {{2}} = \dfrac{4} {{2}} $ Simplify: $ |x - 4| = 2$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 4 = -2 $ or $ x - 4 = 2 $ Solve for the solution where $x - 4$ is negative: $ x - 4 = -2 $ Add ${4}$ to both sides: $ \begin{eqnarray} x - 4 &=& -2 \\ \\ {+ 4} && {+ 4} \\ \\ x &=& -2 + 4 \end{eqnarray} $ $ x = 2 $ Then calculate the solution where $x - 4$ is positive: $ x - 4 = 2 $ Add ${4}$ to both sides: $ \begin{eqnarray} x - 4 &=& 2 \\ \\ {+ 4} && {+ 4} \\ \\ x &=& 2 + 4 \end{eqnarray} $ $ x = 6 $ Thus, the correct answer is $x = 2 $ or $x = 6 $.